In Problem B1, it's easy to identify the most and least fair allocations. It's more difficult to decide the degree of fairness among the remaining three allocations. Note 3

We need a more objective way to measure how close an allocation is to the fair allocation. Here's one method:

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Arrange the stacks in the allocation in increasing order.

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Move 1 coin from an above-average stack to a below-average stack. (Here, "average" refers to the mean.) This move will create a new allocation of the coins with the same mean.

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Continue moving coins, one at a time, from an above-average stack to a below-average stack until (if possible) each stack contains the same number of coins.

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One measure of the degree of fairness for an allocation is the number of coins you had to move. The smaller this number is, the closer the allocation is to the fair allocation.

Problem B2

Below are three allocations of 45 coins in 9 stacks. For each allocation, find the minimum number of moves required to change the allocation into a fair allocation (i.e., one with 5 coins in each stack).

Allocation A

Allocation B

Allocation C

Problem B3

Based on the minimum number of moves, which of the allocations in Problem B2 is the most fair? Which is the least fair?